Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues

TL;DR

This paper proves that the edge fluctuations of the polynuclear growth model with flat initial conditions converge to a Pfaffian point process identical to the GOE edge scaling, highlighting universality in the KPZ class.

Contribution

It establishes the edge scaling limit of the multilayer PNG model as a Pfaffian process matching GOE eigenvalue statistics.

Findings

Edge of the point process converges to a Pfaffian process

Limit matches the GOE edge eigenvalue distribution

Supports universality in KPZ class growth models

Abstract

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. Through the Robinson-Schensted-Knuth (RSK) construction, one obtains the multilayer PNG model, which consists of a stack of non-intersecting lines, the top one being the PNG height. The statistics of the lines is translation invariant and at a fixed position the lines define a point process. We prove that for large times the edge of this point process, suitably scaled, has a limit. This limit is a Pfaffian point process and identical to the one obtained from the edge scaling of Gaussian orthogonal ensemble (GOE) of random matrices. Our results give further insight to the universality structure within the KPZ class of 1+1 dimensional growth models.